I am using Markov transition probability matrices in my work.
I saw that one can introduce an error term in the calculation of the
probabilities if we know that the inputs used to compute the probabilty
matrices
have some uncertainty.

I saw in a book that

P (i,j) corrected for the error in the inputs = P (i,j) * (1-Error)
for i=j

where Error is a number between 0 and 1 to indicate the above mentioned

uncertainty of tte inputs used to compute the transition probability
matrices P

but this expression is only working with i=j (i.e. only for the
diagonal
elements of the matrix)